LiBH4 is a fast ionic conductor in its high-temperature phase, which is stabilized at room temperature by various chemical substitutions, making it a potential solid electrolyte material for Li-ion batteries. Using first-principles variable-cell-shape molecular dynamics simulations, we reproduce the experimentally observed low- and high-temperature structures. Using the height of a structure-factor-like peak as a collective coordinate, we calculate the free energy differences between the two structures as a function of temperature and substitutional ion concentration. We get good agreement with experiment for I, Cl, and Na, and predict that K is even more effective for lowering the critical temperature. Decomposition of the free energy into enthalpy and entropy reveals that the mechanism driving this lowering varies among substitutional elements. Calculating the full free energy, rather than simply the enthalpy, is therefore crucial to understanding how chemical substitution stabilizes the high conductivity phase.